Marilyn & the Goats: A New Solution to an Old Problem

Introduction: In a previous lecture, we gave a New Definition of Probability. In this lecture, we will show how this definition enables us to resolve the massive amount of controversy which surrounds an apparently simple probability puzzle, known as the Monty-Hall problem. The book The Power of Logical Thinking, quotes cognitive psychologist Massimo Piattelli Palmarini: “No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.” Before proceeding to discuss this problem, we review the key features of our new definition:

  1. Probability is a feature of external reality, not an aspect of our ignorance/knowledge.
  2. Probability exists PRIOR to occurrence of random event.
  3. Probability is EXTINGUISHED after event occurs.

We will use this definition to provide a new answer to the Monty Hall Problem. Our definition gives new insights – both a new solution and an explanation of why the problem has caused so much confusion. We now set out the problem.

Let’s Make a Deal: You are the contestant on a game show. You have to choose between three doors. One of the doors has the grand prize of an Automobile, while the other two have Goats behind them. Suppose you choose Door 1, at random. After you have chosen, but before you know whether you have chosen correctly, the host offers you a choice. The host opens door 3, and reveals a goat. He now gives you and OPTION to switch. The question is: SHOULD You SWITCH?

The standard analysis of this problem comes to the conclusion that you should switch. There are two cases where you start out with the goat. In both of these cases you will win the Auto by switching. There is one case where you start out with the Auto. In this case you will lose by switching. So, you have 2/3 chance of winning an Auto by switching, but only a 1/3 chance of winning if you stick with your original choice. We now proceed to explain why this analysis is wrong.

Standard Analysis of Monty Hall Problem

Our temporal probability model tells us that, BEFORE making the choice, you had a probability 1/3 of selecting the right door. After the choice has been made, probability has been extinguished. You either chose the auto or you did not. Furthermore, the Host KNOWS whether or not you have made correct choice. The game should be analyzed as a DETERMINISTIC game of STRATEGY, with asymmetric information. PROBABILITY is NOT INVOLVED in analysis of this game (as we will show).

Why Standard Analysis is wrong: The standard analysis assumes that host follows mechanical rule – he will ALWAYS reveal a goat, and ALWAYS offer a choice. This is nowhere stated in the description of the game. The host does not say anything about whether or not he always shows all contestants a door with a goat behind it. We will now develop the correct strategic game tree under the assumption that the host has two possible strategies. He can either open the door you have chosen to reveal whether or not you have won the prize, OR he can open an alternative door and offer you the option to switch.

Switching Leads to Loss: Analysis of this strategic game is easy. Whenever you choose a goat, the host should open the door and show you the goat. If you choose the automobile, it is a clever strategy on the part of the host to show you a goat, and to try to persuade you to switch. If you believe the standard analysis that switching is better, you will end up losing the auto.

Strategy Tree For Goats

Reality Check: This analysis is aligned with the portrayal of a game show in the movie “Slumdog Millionaire”. The contestant does not know the answer to the crucial Million dollar final question on the quiz show. The  Gameshow host attempts to build trust, in order to deceive hero by giving him the wrong hint about the answer. Commenting on the probability analysis, the original host of the game, Monty Hall, demonstrated his mastery of contestant psychology. He won 8 out of 8 games, by showing the contestant the goat when he chose it, and psyching him into switching when the contestant made the right choice (NY Times: Behind Monty Hall’s Doors Puzzle). After the choice is made, probabilities are extinguished, and the game should be analyzed as a deterministic game of strategy, with asymmetric information.

Pervasive Confusions: Probability experts opine that confusions arise because “Our brains are just not wired to do probability problems very well.” This opinion has emerged from studies of human behavior in situations of uncertainty. In fact, the problem lies in the theories of rational behavior which are used to judge human behavior. These theories, which blindsided economists and investors to the global financial crisis are wrong. Failure of human behavior to match these theories testifies to the wrongness of the theories. Human behavior has evolved to judge probabilities and take actions which lead to highest survival probabilities. One of the key confusions created by probability theories which do not take the temporal nature of probability into account is discussed below.

Confusing Confidence with Probability: Before Occurrence of random event, probabilities are part of external reality. For a person who is IGNORANT about outcome, his KNOWLEDGE about the real world LOOKS like probability. It is just after the event, rather than before the event. Call this state of knowledge “confidence” to highlight its subjective nature. Despite similarity in appearances, confidence is NOT probability. You can BET on probability because external randomness should be same for all. You cannot BET on confidence because event has occurred, others may KNOW it. The enormous amount of confusion which surrounds the interpretation of confidence intervals exists because of failure to differentiate between pre-event probability and post-event confidence.  

Concluding Remarks: In a study by Granberg and Brown (1995), cited in Wikipedia entry on Monty Hall Problem, only 13% of contestants switched, in actual experiments. Intuitive human behavior is smarter than Marilyn Vos Savant, reportedly the smartest human being (highest IQ) on the planet. The recommendation to switch is based on a probability model being applied to a situation where probabilities do not exist.  This kind of confusion is common because standard probability theory models are NOT situated in time. Time flows in ONE direction. In particular, Bayes rule infers P(B|A) from the reverse probability P(A|B). This cannot be valid for events situated in time, since time flows in one direction only. It is valid to condition later events on outcomes of previous events, but one cannot infer probabilities of earlier events conditional on outcomes of future events.

Postscript: A valid criticism that can be made of the analysis above is that I am analyzing the wrong game. The original game is played according to the (admittedly artificial) rules that the host must show a goat, and must offer the contestant a choice. To understand this better, let us imagine a scenario where this is the very first time the game is being played. After you make a choice, the host shows you a goat, and gives you the choice to switch. The analysis which recommends that you should is based on the ASSUMPTION that the host always behaves like this — he always offers contestants a choice to switch, after showing them a goat. You have no warrant to make such an assumption. Next, let us suppose that AFTER you have made a choice, the host says that I am going to show you one of the doors (with goat) and offer you an option to switch. You ask him if he PLANS to offer this choice to all future contestants. He says YES, I will always offer contestants an option to switch. Should his intentions about what he plans to do in the future affect your choice? You are concerned that he is offering you an option only because you have made the right choice, and he wants to direct you away from it. Should his promise to play in a certain way in the future alleviate your suspicion? Next, suppose that the past patterns show that the host has in fact behaved in this way – offered all contestants a choice. But suppose that the host has retired and has been replaced by a new person who is acting as the host of the gameshow. Should you believe that this new host will be following the same mechanical rules followed by the previous host? Suppose that he announces that he plans to make some changes in the game format, to make it more exciting, without explaining what these changes are. Should you suspect that he is planning to save the show some money by offering choices ONLY to contestants who have made the right choice? Next suppose it is the same host, with a history of offering choices to all contestants. But at the start of the show he announces that he plans to make changes in the show format, without specifying what they are. Now how should you behave? All of this shows that the answer to the question depends on psychology and trust, and not upon probability.

The Lady and the Tiger: In fabled lands of yore, the King devises a clever plan to eliminate a commoner who has overstepped class boundaries to fall in love with the princess. Before a huge audience, the king asks the suitor to choose between two doors, one of which contains a hungry tiger, while the other contains a lady. Depending on his choice, the suitor will get eaten, or will wed the lady. In either case, he will be removed as a contender for the hand of the princess. The princess signals the suitor to choose door A. What is the probability that there is a lady behind door A? The answer is that there is no probability involved here. Either there is a lady or there is a tiger. Coming to the confidence issue, the value of the signal provided by the princess involves weighing whether the jealousy of the princess is stronger than her love. Probabilities are not involved in the decision.

4 thoughts on “Marilyn & the Goats: A New Solution to an Old Problem

  1. I and a friend had a dispute about a historical event. He thought the battle of Gettysburg was fought in July. I was sure it was in June. We agreed that we would trust Wilkepedia to be correct and we put a bet on it. Wilkepedia confirmed his view and I paid up. What is impossible about that story? Of course you can bet on the past if you agree on an authoritative source of knowledge. Just as you can bet on the future if you can rely on time to tell the answer. You cannot sensibly bet on the remote future, like the Presidential election of 2320, because time won’t tell you in time. That demonstrates that futurity is neither necessary nor sufficient for making bets.
    Your statement that there is such a thing as objective probability is itself a metaphysical statement. We can only talk of probability of events when we don’t know their outcome. Our ignorance is a necessary condition for talking of probabilities. There is no particular basis for projecting our uncertainty on to the universe. Things may all be entirely determinate for all we know. Perhaps, perhaps not. What is the probability that probability is “objective”? Remember Occam: don’t multiply unnecessary entities. Epistemological uncertainty hence probability is all we have to be concerned with.
    Your Marilyn and goats story shows you get different results if you change the rules of the game. True and fair enough but it tells you strictly nothing about probability theory.

  2. I am concerned with defining probability, not with solving puzzles, The key assertion is that it is crucial to differentiate it across time – pre-occurrence and post occurrence.

  3. Given that you (@ghholtham) have made sharp and perceptive comments in the past, I took your failure to understand as a signal that my writing was not clear enough. I have now modified the POSTSCRIPT in an attempt to provide greater clarity.

  4. This is such an important topic!
    For instance I find in my work that so many people come to accept a model as a truth machine rather than a tool for thought. Too many people, many of them very bright mathematicians, can’t distinguish between the real world and the abstract world of mathematics.
    E.g. the textbook question of a fair coin that comes up heads 99 times in a row … what is the probability of head on the next toss? The classic answer is 50:50, but this must be wrong because the chance of 99 heads is so infinitesimal that the initial statement about the coin must be a lie.
    I think this should be a standard part of teaching on probability, that bluffers and cheats abound in the world and that we should all beware.

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